Electric Potential, Electric Energy, & Capacitance

Gravitational and Electric Potential: Analogies

The concept of potential energy is fundamental in both gravitational and electric fields. In a uniform gravitational field, the gravitational potential energy depends on the mass and elevation, while the gravitational potential depends only on elevation. Similarly, in an electric field, the electric potential depends only on position, while the electric potential energy depends on both the charge and the electric potential.

• Gravitational Potential Energy:

• Gravitational Potential:

• Electric Potential Energy:

• Electric Potential: (measured in volts, V; 1 V = 1 J/C)

• Electric potential is a scalar quantity.

• Potential energy depends on the object (mass or charge), while potential depends only on location.

Example: Two objects at the same elevation have the same gravitational potential, but if their masses differ, their gravitational potential energies differ. Similarly, two charges at the same location in an electric field have the same electric potential, but their electric potential energies differ if their charges are different.

Electric Fields, Electric Potential, and Work

Electric fields are conservative, meaning potential energy can be defined. Only changes in electric potential and potential energy are measurable. The sign of the charge affects how potential energy changes as the charge moves in the field.

• Work in Electric Field:

• For a uniform electric field: and

• Assuming :

• Change in potential:

Objects move from higher to lower potential energy in any field. For positive charges, high electric potential means high potential energy; for negative charges, the opposite is true.

Electric Field and Potential from Point Charges

The electric field and potential from point charges are described by specific equations. The field is related to force, and the potential is related to potential energy.

• Electric Field: (general), (point charge)

• Electric Potential: (general), (point charge)

• Potential Energy Between Two Point Charges:

Example: The potential at a point due to multiple charges is the sum of the potentials from each charge.

Capacitors and Capacitance

A capacitor is a device made of two conductors separated by a dielectric. It stores electric energy and its capacitance depends on its physical characteristics.

• Capacitance:

• Charge-Voltage Relationship:

• SI unit for capacitance: Farad (F), where

• Capacitance changes only if the physical characteristics change.

• Voltage remains constant if the capacitor is attached to a battery.

• Charge remains constant if the capacitor is isolated.

Example: For a parallel plate capacitor, if the plate separation is halved, the capacitance doubles.

Dielectrics in Capacitors

Dielectrics increase the capacitance by lowering the electric field for the same amount of charge. The dielectric constant quantifies this effect.

• Dielectric Constant:

• Capacitance with Dielectric:

• Inserting a dielectric increases capacitance and can affect charge and voltage depending on whether the capacitor is isolated or connected to a battery.

Energy Stored in Capacitors

Capacitors store energy, which can be calculated in several ways. Energy density is the energy per unit volume.

• Energy Stored:

• Energy Density:

Example: If a dielectric is inserted while the voltage remains constant, the energy stored increases by a factor of .

Summary and Review

• Fields and forces are vectors; potential and potential energy are scalars.

• Key relationships: , , (uniform field), , , , (point charges)

• Capacitors: ,

Additional info: The images provided visually reinforce the concepts of electric potential (lightning) and gravitational potential energy (mountain elevation).